The proposed research will be on mathematical and statistical models that can be used to used for the analysis of genetic data in human and other populations. Particular emphasis will be placed on accounting for patterns on variation among individuals within populations and variation among different populations. The overall goal of this research is the development of methods for using patterns of variation to understand the history of human and other populations and to assist with efforts to identify genes affecting genetic diseases. The history of population growth, dispersal and intermixture with other populations is reflected in patterns of genetic variation across the genome. The history of an individual mutation is reflected in patterns of variation at genetic loci that are closely linked to that mutation. The specific goals are (1) to model the history of individual mutations causing genetic diseases in order to understand where and when they arose and whether carriers of those mutations have an advantage survival and reproduction, (2) to make better use of non-random associations of alleles at different genetic loci (called linkage disequilibrium) to locate gene causing genetic diseases, (3) to model the effects of natural selection that acts to maintain genetic diversity at genetic loci such as the loci in the major histocompatibility (MHC) loci in humans and other species, and (4) to develop theories that relate the risk of a genetic disease in close relatives of an individual who has the disease to the number of genetic loci affecting the disease and to the kinds of interactions among those loci. The mathematical methods used in this research will be primarily from the theory of probability and stochastic processes. Analytic results will be obtained whenever possible and will be supplemented by computer simulations. Methods of data analysis will be tested both with simulated and real data. Computer programs that are found to be useful for data analysis will be made freely available.